Dodgson's Eight - A Heap of Sorites
This concludes my analysis of the hitherto unsolved Eight Problems from Part II in the appendix to Dodgson's Symbolic Logic. Links to all the articles are collected here for easy reference:
The purpose of these articles is to provide the solutions for the first time, and to demonstrate that a relatively simple, yet powerful, boolean algebra is capable of solving these advanced problems easily. In the process, it was discovered that the premise set of Problem 2 was redundant, that Dodgson begged the question in Problem 7, and that the nursery rhyme is not a vallid syllogism. Problems 6 and 8 are, by far, the most difficult ones.
My strange journey to these topics began in 1972 with the acquisition of Laws of Form by G. Spencer Brown. The book is a very difficult read, and I do not recommend it - Brown's literary style is eclectic and obscure, his unusual notation and interpretation are deficient, and he has a decidedly mixed reputation. However, he suggested a very simple method for solving Dodgson's last example sorites (No. 60, p. 124) in Appendix 2, Interpretative Theorem 1, p. 123, which caught my interest and led me to read and re-read the book over a period of several years, trying to understand it. [Note: I am an Electrical Engineer with training in Boolean algebra, logic design, and symbolic logic, and still I found the book exceptionally difficult.] Eventually, I reformulated Brown's system into a more conventional notation while retaining Brown's basic structure and axioms/initials. In the process of studying Aristotlean Logic and Syllogism, I discovered the proper form (Syllogistic Theorem below) for Brown's Interpretative Theorem 1 which did not have Brown's difficulties with invalid syllogisms (which he tried unsucessfully to explain away), and the proper form for the Existential Syllogisms (see below), making all 24 valid Aristotlean syllogisms substitutional instances of only two fundamental forms. As mentioned in Problem 2, I first encountered Dodgson's Eight Problems on a blog, applied my newly discovered methods to it, solved it in 30 minutes, asked the blog writer if the answer was correct, and was surprised to learn no one apparently knew the answers to these eight problems.
For those wishing to follow the calculations in the analysis of Dodgson' Eight Problems, I here present that part of my formal system necessary for understanding. There are only two operations: disjunction, indicated by the concatenation of symbols, and complementation, indicated by a postfix tilde "~".
|The binary logical operations are defined as:|
[X1 ∨ X2] = [X1 + X2] = X1X2 Disjunction (Or)
[X1 ∧ X2] = [X1 · X2] = (X1~X2~)~ Conjunction (And)
[X1 ⇒ X2] = X1~X2 Material Implication
[X1 - X2] = (X1~X2)~ Difference
[X1 ≡ X2] = (X1X2)~(X1~X2~)~ Material Equivalence
[X1 ^ X2] = [X1 ⊕ X2] = (X1~X2)~(X1X2~)~ Exclusive Or
[X1 | X2] = X1~X2~ Stroke (Nand)
|For all substitutions of real variables R1, R2 and R3 :|
a) ρ(R1R2)~ρ(R2~R3)~ρ(R1R3) = U or, written as an implication:
ρ(R1R2) · ρ(R2~R3) ⇒ ρ(R1R3) Hypothetical Syllogism
and, by appropriate substitution and re-arrangement, the corollary:
ρ(R1R2) · ρ(R2R3)~ ⇒ ρ(R1~R3)~
b) ρ(R1R2)~ρ(R2R3)~ρ(R1~R3~)~ρ(R2) = U or, written as an implication:
ρ(R1R2) · ρ(R2R3) · ρ(R2)~ ⇒ ρ(R1~R3~)~ Existential Syllogism
|The propositional operator ρ(...) is defined as ρ(R) = U if R = U, and ρ(R) = U~ otherwise, converting the class or set content of the distinction R to a binary propositional value.|
This theorem is the tool which was used in the solution of Dodgson's eight problems (with the exception of Problem 6) - its repeated application eliminates all terms which appear both complemented and uncomplemented. The interested reader should now be able to follow, in detail, the calculations in the eight articles on Dodgson's sorites.
Dodgson had three such rules (Fig. I, II, and III here), but his notation was so peculiar that he failed to realize that his Fig. I and II were equivalent. Brown, on the other hand, failed to explain the Existential Syllogisms at all, and had difficulties with invalid syllogisms because of his defective formulation and interpretation, confusing classes and propositions..